That fitting you’re referring to in your question is most likely an Air Admittance Valve (AAV). This type of venting is rarely approved by plumbing inspectors, it is however the easiest method to vent a kitchen island sink. Since air admittance valve is a mechanical device it sometimes fails and starts releasing sewer gases under the sink.
You can replace it with a new one and see if it helps, this should be a matter of unscrewing it from the pipe. The most popular brands are Oatey and Studor, they are more expensive but maybe they’ll give you better performance…
There are other, more conventional ways to vent a kitchen island sink as well, but you would probably need a plumber to look into it.
Let me know if there’s anything else I can help you with. Just click on "comment" below.